Gauge Amplitude Identities by On-shell Recursion Relation in S-matrix Program

TL;DR

This work demonstrates that four key color-ordered gauge-theory amplitude relations—color-order reversed, U(1) decoupling, KK, and BCJ—can be derived purely from on-shell recursion (BCFW) within the S-matrix framework, without reference to a Lagrangian. It provides the first pure field-theory proof of the BCJ relation by showing that all nontrivial identities follow from a fundamental BCJ relation via induction plus the KK and U(1) decoupling constraints. The method unifies these identities under on-shell techniques and reveals a deeper connection to gravity through the observed bonus $1/z^2$ behavior. The results have implications for amplitude computations, loop extensions, and the gauge–gravity correspondence via KLT relations.

Abstract

Using only the Britto-Cachazo-Feng-Witten(BCFW) on-shell recursion relation we prove color-order reversed relation, $U(1)$-decoupling relation, Kleiss-Kuijf(KK) relation and Bern-Carrasco-Johansson(BCJ) relation for color-ordered gauge amplitude in the framework of S-matrix program without relying on Lagrangian description. Our derivation is the first pure field theory proof of the new discovered BCJ identity, which substantially reduces the color ordered basis from $(n-2)!$ to $(n-3)!$. Our proof gives also its physical interpretation as the mysterious bonus relation with ${1\over z^2}$ behavior under suitable on-shell deformation for no adjacent pair.